geometry problems from Metrix Mathematical Olympiad 2020 and the Shortlist (unused problems) with aops links
2020
first posted here
collected inside aops here
contest problems
Does there exist a heptagon $P$ satisfying the following:
There does not exist a square with exactly three of the vertices on the boundary of $P$.
by MNJ2357
$ABC$ be a scalene triangle. Let $D$ be a point on $\overline{BC}$ such that $\overline{AD}$ is the angle bisector of $\angle BAC$. Let $M$ be the midpoint of $\widehat{BAC}$ of the circumcircle of $\odot(ABC)$. Let $\overline{DN}\perp\overline{BC}$ where $N$ lies on $\overline{AM}$. $K$ be a point on segment $NM$ such that $3KM=KN$. Let $P,Q$ be two points on $\overline{AM}$ such that $KP=KQ$ and $BCPQ$ lie on a circle $\omega$. Show that the reflection of $D$ over $A$ lies on $\omega$.
by Amar_04
sample problems
An acute triangle $ABC$ ($AB\neq AC$) with circumcircle $\omega$ is given, and $M$ is the midpoint of arc BAC . Suppose $D$ is the point on segment $BC$ such that the circumcircle of $MAD$ is tangent to $BC$, and suppose $P$ is the point on segment $BC$ such that $AP$ and $MD$ concur on $\omega$. Let $Q$ be the intersection of $MA$ and $BC$, $N$ be the midpoint of $MD$, and $X$ be the intersection of $NQ$ and $MP$. Prove that $\angle MDB=\angle XDC$.
by MNJ2357
The point $H$ is a orthocenter of the triangle $ABC$ , given the conditions that $\odot(ABH)\cap HC=T,\odot(ACH)\cap HB=K$, $\odot(KTA)\cap \odot(ABC)=N$ , $AH\cap \odot(HTK)=P,\odot(HTK)\cap \odot(PNA)=Q$
$\odot(HBC)\cap HQ=M$ with $HACB'$-cyclic, $HABC'$-cyclic. $\angle BAH=\angle CC'A,\angle CAH=\angle BB'A$
$\odot(ABB')\cap \odot(ACC')=H'$ .Then prove that $H',B',C',M$-cyclic
by Functional_equation
unused problems
Let $ABC$ be a triangle with centroid $G$. $GD$, $GE$, $GF$ are symmedians of triangles $GBC$, $GCA$, $GAB$, respectively. $L$ is symmedian point of $ABC$. $L^*$ is inversion of $L$ in circle $(ABC)$. Prove that $AD$, $BE$, $CF$ and $GL^*$ are concurrent.
by buratinogiggle
Let the points $H$ and $N_9$ to be the orthocenter and the Nine-Point center of $\vartriangle ABC$ respectively and let $R$ to be the circumradius of $\vartriangle ABC$. If $\angle BAC = \alpha$ , $\angle ABC = \beta$, $\angle ACB = \gamma$. Prove that$$\left( \frac{2HN_9}{R}\right)^2 \le 9 - 8\sqrt3 \sin \alpha \cdot \sin \beta \cdot \sin \gamma.$$
by Functional_equation
Let $D, E,$ and $F$ be the respective feet of the $A, B,$ and $C$ altitudes in $\triangle ABC$, and let $M$ and $N$ be the respecive midpoints of $\overline{AC}$ and $\overline{AB}$. Lines $DF$ and $DE$ intersect the line through $A$ parallel to $BC$ at $X$ and $Y$, respectively. Lines $MX$ and $YN$ intersect at $Z$. Prove that the circumcircles of $\triangle EFZ$ and $\triangle XYZ$ are tangent.
by mathman3880
In triangle $ABC$ denote $I_A$ as the center of the excircle wrt $A$. Let the excircle touch the sides $AB, BC, CA$ at $M, N, P$ respectivly. Let $T$ be the midpoint of $MP$ , $TC$ and $MN$ hit at $T'$ and $BC$ , $PM$ hit at $T''$ . Prove $I_A$ is the orthocenter of $TT'T''.$
by Mr.C
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